{
  "id": "2401.14301",
  "version": "v1",
  "published": "2024-01-25T16:50:35.000Z",
  "updated": "2024-01-25T16:50:35.000Z",
  "title": "Some determinants involving quadratic residues modulo primes",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\\equiv3\\pmod4$ and $a,b\\in\\mathbb Z$ with $p\\nmid ab$, we prove that $$\\det\\left[1+\\tan\\pi\\frac{aj^2+bk^2}p\\right]_{1\\le j,k\\le\\frac{p-1}2}=\\begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p-3)/4}&\\text{if}\\ (\\frac{ab}p)=-1,\\end{cases}$$ where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-25T16:50:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A15",
      "11C20",
      "15A15",
      "33B10"
    ],
    "keywords": [
      "quadratic residues modulo primes",
      "determinants",
      "legendre symbol",
      "conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}